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# Cauchyâ€™s Mean Value Theorem

Cauchyâ€™s mean value theorem is a generalization of the normal mean value theorem. This theorem is also known as the Extended or Second Mean Value Theorem. The normal mean value theorem describes that if a function f (x) is continuous in a close interval [a, b] where (aâ‰¤x â‰¤b) and differentiable in the open interval [a, b] where (a < x< b), then there is at least one point x = c on this interval, given as

**f(b) â€“ f (a) = fâ€™ (c) (b-a)**

It establishes the relationship between the derivatives of two functions and changes in these functions on a finite interval.

Letâ€™s consider the function f(x) and g(x) be continuous on an interval [a,b], differentiable on (a,b), and g'(x) is not equal to 0 for all x Îµ (a,b). Then there is a point x = c in this interval given as

## Proof of Cauchyâ€™s mean value theorem

Here, the denominator in the left side of the Cauchy formula is not zero: g(b)-g(a) â‰ 0. If g(b) = g(a), then by Rolleâ€™s theorem, there is a point d ? (a,b), in which g'(d) = 0. Therefore, contradicts the hypothesis that g'(x) â‰ 0 for all x ? (a,b).

Now, we apply the auxiliary function.

F (x) = f (x) + Î»g(x)

And select Î» in such a way to satisfy the given condition

F (a) = f (b). we get,

f (a) + Î»g(a) = f (b) + Î»g(b)

= f(b)-f (a) = Î»[g(a)- g(b)]

And the function F (x) exists in the form

The function F (x) is continuous in the closed interval (aâ‰¤x â‰¤b), differentiable in the open interval (a < x< b) and takes equal vales at the endpoints of the interval. So, it satisfies all the conditions of Rolleâ€™s theorem. Then, there is a point c exist in the interval (a,b) given as

Fâ€™ (c) = 0.

It follows that

By putting g (x) = x in the given formula, we get the Lagrange formula:

Cauchyâ€™s mean value theorem has the given geometric meaning. Consider the parametric equations give a curve ? X = f (t) and Y = g (t), where the parameter t lies in the interval [a,b].

When we change the parameter t, the point of the curve in the given figure runs from A (f (a). g(a) to B (f(b), g (b)).

According to Cauchyâ€™s mean value theorem, there is a point (f(c), g(c)) on the curve ? where the tangent is parallel to the chord linking the two ends A and B of the curve.

## Questions Based on Cauchyâ€™s mean value theorem

Question 1:

Calculate the value of x, which satisfies the Mean Value Theorem for the following function

F(x) = x^{2} + 2x + 2

Explanation:

Given

f(x) = x^{2} + 2x + 2

According to Mean Value theorem,

2c = -7

C = -7/2

Question 2:

Calculate the value of x which satisfies the Mean Value Theorem for the following function

F(x) = x^{2} + 4x + 7

Explanation:

Given

f(x) = x^{2} + 4x + 7

According to Mean Value theorem,